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Journal of Physics A Mathematical and General

Remarks on the conventional formulation for interaction of radiation with matter

Qian Shang-Wu

Published Feb 21, 1987• 2 citations
Publisher

Abstract

The conventional formulation for interaction of radiation with matter uses a zero potential Hamiltonian H 0 as an unperturbed Hamiltonian and this procedure leads to a conflict between the gauge invariance of probability amplitudes and the Born rule. It is pointed out that if one uses a dark Hamiltonian H d as the unperturbed Hamiltonian and considers its gauge dependence, then this conflict no longer exists. From this point of view the author proves that the Lamb gauge and the Coulomb gauge in the electric dipole approximation give the same gauge-independent probability amplitudes, and thus the A.p against r.E controversy becomes unimportant.

DOI
10.1088/0305-4470/20/3/023

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The conventional formulation for interaction of radiation with matter uses a zero potential Hamiltonian H 0 as an unperturbed Hamiltonian and this procedure leads to a conflict between the gauge invariance of probability amplitudes and the Born rule. It is pointed out that if one uses a dark Hamiltonian H d as the unperturbed Hamiltonian and considers its gauge dependence, then this conflict no longer exists. From this point of view the author proves that the Lamb gauge and the Coulomb gauge in the electric dipole approximation give the same gauge-independent probability amplitudes, and thus the A.p against r.E controversy becomes unimportant.
The conventional perturbation theory in quantum mechanics for the interaction of electromagnetic radiation with matter, which is based on the time-dependent vector and scalar potentials, is shown to be dependent on the gauge. The problem is reformulated, using the approach of Yang, in a gauge-invariant way, in which the transitions between states are induced by the quantum-mechanical power operator. In order to use an unperturbed Hamiltonian which does not involve the electromagnetic field, the gauge is chosen to give the electric dipole interaction. This choice can be made if the magnetic field is negligible and the electric field is slowly varying over atomic dimensions. For a comparison of the gauge-invariant and conventional formulation, a two-level atom in a radiation field is considered. For the probability that an atom originally in the lower state will make a transition to the upper state when the field is not in resonance, the gauge-invariant formulation gives the correct result, whereas the gauge-dependent formulation gives an incorrect result.
The case of a single charged particle with potential energy is considered when the classical electromagnetic field is zero. The vector and scalar potentials describing this situation are not necessarily zero. When the conventional approach to radiation problems using the interaction A⋅p plus p⋅A plus A2 plus A0 (constants omitted) is used, time-dependent probabilities are predicted if the vector potential A=−∇Λ≠0 even though the electromagnetic field is zero. The reason for this unphysical behavior is that eigenfunctions of the unperturbed Hamiltonian, which is not gauge invariant, are used to calculate the probability amplitudes. When a manifestly gauge-invariant procedure is applied to this case the probabilities are constant in all gauges. The reason that physical results are obtained is that eigenstates of the energy operator are used to calculate the probability amplitudes.
The gauge-invariant formulation of quantum mechanics is applied to the charged harmonic oscillator in an electromagnetic field in the electric dipole approximation to obtain the probability that the oscillator is in a particular state at time t. This probability is compared with the corresponding 'probability' calculated from the conventional approach to the interaction of radiation with matter using the interaction A.p. The probabilities do not agree with each other. From the principle of gauge invariance it is concluded that the probability calculated from the gauge-invariant formulation is correct in this case, and that the conventional approach in general is incorrect.
It is pointed ont that there are two kinds of methods, to construct gauge inde-pendent transition probabilities (GITP), one kind corresponds to active view of gauge invariance in quantum mechanics, another kind corresponds to the passive view. Two kinds of view are interchangeable by means of unitary transformation. From the active view, all known methods of constructing GITP consist in choosing a gauge dependent unitary operator which transforms the non-physical quantity H0 into a true physical quantity to be measured, where H0 is the unperturbed Hamiltonian in the conventional formulation for interaction of radiation with matter.
The space-translation method for treating the interaction of electromagnetic radiation and matter in the electric dipole approximation is investigated from the standpoint of gauge invariance. Unless the wavefunction of the space-translation method is properly transformed, the probability of finding the system in an energy eigenstate is not in general correct. In the electric dipole approximation the probability of finding the system in an energy eigenstate is most conveniently calculated in the electric field gauge where the interaction is of the length form. The length form of the interaction is not in general equal to the velocity form or the acceleration form which comes from the space-translation method. The charged harmonic oscillator is used as an example.
A recent investigation of the interaction between the Maxwell radiation field and neutral atoms or molecules has led us to consider the relation between the straightforward interaction Hamiltonian between particle (k) and the radiation field, and the more physical, and explicitly gauge invariant, couplings of the form q. E and µ. H, etc., where q and µ are the electric and magnetic dipole moments of a system of particles. That these latter form of interaction energies follow from well-formulated approximations for atoms in external fields has been known for many years (Goeppert-Mayer(1); see also Rosenfeld(2)). When the Maxwell field is an internal and dynamical field further discussion is necessary, especially concerning the Coulomb interactions. Their advantage has recently been shown in dealing with dipole interactions in the long range Van der Waals force between two neutral systems where the complete coupling, in dipole approximation, is given by — q.ET for each neutral system, where ET is the transverse electric field. The short-range London limit is contained in this coupling despite the absence of longitudinal electric fields (Power and Zienau (3)). In this note we extend these arguments to higher electric multipoles and magnetic multipoles. We also demonstrate their use in the quantum theory of scattering of light, and in particular the derivations of Stephen (4) of the phase angle between emergent beams in optical double refraction phenomena are shown to follow immediately.